Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 733 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 35\cdot 733 + 97\cdot 733^{2} + 279\cdot 733^{3} + 74\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 109 + 115\cdot 733 + 323\cdot 733^{2} + 85\cdot 733^{3} + 250\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 126 + 330\cdot 733 + 150\cdot 733^{2} + 218\cdot 733^{3} + 125\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 257 + 171\cdot 733 + 367\cdot 733^{2} + 68\cdot 733^{3} + 60\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 316 + 278\cdot 733 + 487\cdot 733^{2} + 330\cdot 733^{3} + 627\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 398 + 434\cdot 733 + 137\cdot 733^{2} + 359\cdot 733^{3} + 461\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 488 + 252\cdot 733 + 162\cdot 733^{2} + 150\cdot 733^{3} + 283\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 496 + 581\cdot 733 + 473\cdot 733^{2} + 707\cdot 733^{3} + 316\cdot 733^{4} +O\left(733^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,5)(2,6)(3,4)(7,8)$ |
| $(1,2)(3,7)$ |
| $(1,3,2,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-4$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $0$ |
| $4$ | $4$ | $(1,5,3,8)(2,6,7,4)$ | $0$ |
| $4$ | $4$ | $(1,8,3,5)(2,4,7,6)$ | $0$ |
| $4$ | $4$ | $(1,3,2,7)(4,8)$ | $0$ |
| $4$ | $4$ | $(1,7,2,3)(4,8)$ | $0$ |
| $4$ | $4$ | $(1,8,2,4)(3,6,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
